Hydrodynamic loading on multi component bodies

TECHNISCHE UNIVERSII'EIT

Laboratodum voor

Scheepshydromechaj

-kchlef-Meketweg 2,2628 CD

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T&. O15-1IN1

_{Fac 015-781833}

YDR0DYNAMIC LOADING ON MULTI-COMPONENT BODIES

R. Eatock Taylor J. Zietsman

University College London University College London

SUMMARY

The problem of predicting wave induced reaponses of complex structures such as certain tethered buoyant platforms and wave energy devices is discussed. Emphasis is placed on the

requirement to achieve economically sufficient accuracy in the wave frequency response analysis

to enable wave drift effects to be reliably estimated.

A method of analysis meeting this requirement is first briefly described. This coupled

element formulation is based on combining a finite element idealisation of the fluid region close to a body with a boundary integral representation f the far field behaviour. It is readily

adapted to multi-component bodies. Results obtained for several such systems are extensively

discussed, including a semisubmersible catamaran and two closely spaced- floating bodies free to

move independently. It is shown that the complex three dimensional hydrodynamic interaction

between such bodies (including standing wave eacts) cay be accurately predicted by this

economi-cal numerieconomi-cal technique. Finely resolved pressure distributions on submerged surfaces, and

details of fluid particle kinematics, may also be estimated.

t.,.

iI;

X(V fC

INTRODUCTION

CI31

For sever.ear's rical methods have been available for evaluating the wave diffraction

loads on large vIuth fixd offshore structures such as storage tanks and gravity platforms.

Ex-tensions have been made to permit computation of hydrodynamic loads (added mass and damping effects), and the resulting wave frecuency responses, of freely floating and partially restrained submerged large bodies such as barges, certain tethered buoyant latforms and wave energy devices.

More recently these techniques have been further extended in order that mean and slowly varying wave drift resonses might be assessed. For a large group of examples, satisfactory agreement

has been achieved between forces and responses computed by these techniques and corresponding

quantities measured on models in regular and random waves. A list of typical cases for which

comparisons have been made has been given by the ITTC (1981), and many other unpublished data exist indicating the apparent success of the numerical methods.

It has also become clear in recent years, however, that the available methods, while highly satisfactory for some large bodies of relatively simple geometry, are considerably less effective

in other cases. This is particularly evident where the geometry is complex, and interactiot

effects between bodies become important. Stated simply, the main problem is that it becomes

excessively expensive to achieve the numerical accuracy necessary to resolve satisfactorily these

complex flows. The difficulty is highlighted when wave drift forces are required, since these

are second order quantities depending on the difference between first order effects (which

therefore must be accurately evaluated). For multi-body problems this limitation can be

parti-cularly severe, since drift forces are more sensitive to interaction effects than first order

wave forces.

Most existing numerical techniques for solving these problems are based on discretisation of a boundary integral equation on or near the submerged surface of the body. Source distribution

methods are typical of this approach to wave diffraction analysis. Several computer programs based on this approach are now commercially available, some of which are listed in the

afore-mentioned ITTC Report. To the best of our knowledge these boundary integral procedures are all

based on an assumed constant distribution of the governing variable (source strength, velocity potential etc.) over each panel, facet or element into which the surface is divided. It would

therefore not be surprising if large numbers of panels were required for the accurate resolution

of complex wave flows. But this potential economic limitation is not the only difficulty

asso-ciated with these procedures. It is well known that boundary integral equations are often subject

to the phenomenon of "irregular frequencies", near which the numerical procedure breaks down due

to poor numerical conditioning. In many practical cases the lowest irregular frequency falls

above the

frequency

range of engineering interest, or its effect can be mitigated by the simple

expedient of interpolation between well-conditioned results. In cases of certain unconventional

geometries, however, or where higher frequency dynamic response is of interest fora fatigue analysis, irregular frequencies can pose a significant problem.

Another group of numerical techniques for wave d''-ction and radiation problems is that based on the localised finite element method, which does not manifest irregular frequencies. The

approach has been expounded by Chen and Nei (1974) and Bai and Yeung (1974), with subsequent

con-tributions by Euvrard et al (1977) and others. The finite element discretisation in an inner

dcmain surrounding the body is coupled with a series solution in the outer domain, extending to

infinity in the horizontal direction. The method can be particularly efficient for complex

geometries in relatively shallow water depths, but since the element mesh must be taken down to the seabed (even for a floating body) this advantage is lost in deep water. Perhaps for this

reason, coupled both with the significant investment in preparing the element mesh in the bsence

of a satisfactory mesh

generator

and with the relative lack of widespread knowledge about this

approach, the localised finite element method doca not appear yet to have been widely used by

the offshore industry.

A third approach, more recently developed, combines many of the advantages of the finite

element and boundary integral methods. Finite elements are again used in the

inner

domain, but

coupling with the infinite outer domain is achieved by a boundary integrai discretisation on the

fictitious surface bounding the inner element region. The basis of the method was suggested by

Zienkiewicz et al (1977). Its application t' hydrodynamic analysis, and its development to permit

the fictitious surface to be lifted off the sea bottom (thereby reducing the inner element region) havebeen described in Latock Taylor and Zietsman (1981a, 1981b - referred to n the following

as A and B respectively) . A somewhat different implementation has also been given by Lenoir and

,Iami (1978). A two

dimensional

formulatiofl was discussed in some detail in A, where extensive

comparison was made with analytical solutions and with numerical results from the alternative

localised finite element (boundary series) method.

Consideration

was given to the well-known

hydrodynamic reciprocal relations and to the question of irregular frequencies associated with

the boundary integral idealisation. A brief introduction to the three dimensional formulation

was presented in B, together with limited comparison of resulta with analytical solutions and

sperimental data, for simple geometries.

with 426 Here n; =

-)0/n)

so n1, n2, n3 $ (S) dS,

The present paper demonstrates that hydrodynamic loading and resulting responses for highly complex three dimensional problems may be efficiently computed by the coupled finite element-boundary integral ("coupled element") approach described in A and B. The formulation is first

briefly reviewed, with a discussion of features contributing to efficiency such as explicit integration, choice of element mesh, symmetry, and the method of equation solution. The majority

of the paper, however, is concerned with application of the coupled element approach, in order to illustrate several features of the flow past complex bodies. These include three dimensional

effects and interactions between vertical and horizontal floating cylinders, as well as other

groups of bodies. As a result of comparisons with known solutions, and convergence studies using

progressively finer meshes, it is believed that the results presented here exhibit a high degree

of accuracy, such as is required in wave drift calculations. This is shown to be achievable by

relatively economical computations. The paper also demonstrates that the numerical phenomenon

of irregular frequencies - to which the conventional boundary integral methods are susceptible

-may be satisfactorily circumvented, so that there is unlikely to be any confusion with the physi-cal standing wave effects present in multibody problems.

REVIEW OF COUPLED ELEMENT FORMULATION

Governing Equations

Full details of the mathematical background are given by Zietaman (1982 - referred to

henceforward as Z) . The analysis is based on the assumptions of ideal flow. Velocity potentials

are defined for motions at frequency u, such that

= Re [

etJ

We select a Cartesian coordinate system Oxyz located in the mean free surface with Oz positive

upwards. The linearised free surface (SF) and bottom (S5) boundary conditions are then

on z = -d (3)

and

g-4.

on z0

(4

respectively. We may read the flow as a superposition of incident, diffracted and radiated

waves, with potentials - . The incident wave is given by r 0, the radiated waves

correspon-ding to rigid body motions of a single body are associated with r = 1 to 6, and the diffracted

wave is designated by r = 7. The diffraction and radiation potentials satisfy the radiation

condition at infinity (e.g. Newman 1977), and the boundary conditions on the boundary surface S0:

(r)

- -ianr on SO (5)

correpond to Cartesian components of the normal vector into the body surface; and n4, n5, n6 correspond to the components of the croas product between this vector and the vector joining the

point on S0 to the point which rotations of the body are spec if ìed.

Hydrodynamic loads are obtained by integrating fluid pressures over the surface,subrzerged

in the fluid of density p. For the linearisd problem this leads to generalised wave forces

f iao $

(0) 7o]

n dS, r = i to 6

r r

so

and added mass and damping matrices, j and X _{respectively, given by}

rs rs

r, s = i to 6

These are used with Newton's Second Law in the formulation of equations of motion for the bodies

considered below. Where there is more than cne body, moving independently, the number of modes

(6)

(7)

X

-rs rs

n, radiation potentials etc. becomes the total number of independent degrees cf freedom,

and S becomes the total submerged surface (although n' may be zero on all but one body for

certain modes). Formulation of the equations of motion for multiple independent bodies, including

the appropriate hydrodynamic interaction terms, has also been discussed by van Oortmerssen (1979).

The coupled element formulation provides an effective numerical solution to the boundary

value problem for (r) The semi-infinite fluid domain is divided into the two regions shown

in Fig 1. The inner region V1 completely enclosing the surface ü of the body or bodies is

idealised by finite elements, and separated from the outer region V2, by a surface SJ. The boundary element idealisation on Sj, involving the classical Green's function for a wave source

in finite depth water, is used to represent the outer region. Continuity across S is ensured

by the following variational procedure (Zienkïewicz et al 1977).

Let be a diffraction or radiation potential in the inner region V1, corresponding to

a fluid velocity n' normal to S0 as given by Eq (5). We define a functional U by

U () =

4 $ (Vó1)2

dv - i

5 dS + iw f n' dS - 4 j

Pi j

dS

Vi _SF

(8)

where SF is that part of the fra! surface associated with V1. Setting to zero the first

varia-tion of 'H with respect to $, yields the governing equavaria-tions, Eq (2), (4) and (5). The

appro-priate behaviour of on S is ensured by enforcing continuity with a potential 2 in the outer

region V2

12

= _{In1 In,} on S

, is required to satisfy the following integral equation for the point P on S whose position

vector is p .4. o 2

() +5

s (q) dS =

f G(;)

I2(q) dS SJ J -4. an _S In

where is the normal out of V2 at the point Q on S whose position vector is . The value of L

is 4

if p corresponds to a node on S at the free surface or bottom of the fluid; otherwise the

value of 1. is 1. o is the solid angle enclosed by S at the point P

(as seen from V2): thus if

(as below) S is a rectangular box, o takes the values 2e, 3 and 3.5e respectively :or points on

a face, an edge or a corner of the box. Specification of G as the wave source Green's function

ensures that the potential 2 satisfies the free surface, bottom and radiation boundary

conditions (John 1950)

Discretisation of Eq (8)-(10) provides a means of approximating the solution for

4,.

On Sj boundary elements are used, with shape functions M interpolating

4,

between nodal values ,.

Equation (10) is therefore represented by

Y 2n = ' ₂

where , and

2n are the vectors containing the terms ,

and 14,/In at node j, and

- 5

GM.dS

1J a i j J 1G V.. = 5. - - _f M. dS 13 0 In j J

G. is-the Green's function associated with node i and the arbitrary point Q (the integration

variable) on S. Within the finite element region, shape functions N are used to interpolate

between nodal values

4,

. Variation of the discretised form of Eq (8) , and substitution of

Eq (11) together with E (9), finally leads-to the equation

(K1 + ) = P

(13)

The matrix elements are defined as follows:

(12e)

(1) Wz K.. _{= $} VN. VN dv - - _j N N. dE y 1 g i D 1 F N3 [W _$

N.M

dS+W

_f N N dS) k=1 kj S3 k

kis

k P.

_t

=

-ia5

N.

_i

n dE

where N3 is the number of nodes on S3 and

W = V

(1 4a

(14b)

(1 4c)

(15)

Numerical considerations governing formulation of these matrices and solution of the

resul-ting equations are discussed in Z. We only note here that both t(') and K2 are symmetric

matrices. The method was originally developed for single body problems, for which V1 isa simply

connected region. For multi-body problems, however, the approach may be greatly improved by

surrounding each body S by a separate region of fluid V!, with associated fictitious boundary s,.

The salient features are summarised below.

Mesh Selection for a Single Fluid Region

Efficient application of this method is dependent on two matters related to mesh geometry. The first concerns the finite elements within the inner region V1, close to the body. They are

required to provide a satisfactory geometric fit to arbitrary submerged surfaces, and to permit an adequate representation of pressure distributions on these surfaces (as obtained from the

velocity potentials). We have selected quadratic isoparametric brick elements (Zjenkiewjcz 1977),

which generally meet both of these criteria. Even more important is the choice of the fictitious

boundary S3 bounding the element region. This has a major influence on the efficiency of the

boundary integral computations and on the ease with which irregular frequencies may be

circum-vented. It is found that a right parsilelipiped (or box) is the optimum choice for analysis of

three dimensional flow past all simple body and some multi-body problems. The aspect ratio of

the box may be chosen arbitrarily. This enables a good fit to be provided for regions enclosing

squat or slender bodies, consistent with avoidance of possible irregular frequencies in the range

of interest, but without excessive distortion of elements. The box also facilitates the use of

cash generation routines for the element region.

The economy of computation associated with the rectangular box specification for Ej derives

from two sources. Firstly, the number of non-zero derivatives of the Green's function G(,) is

minimised, thereby easing the evaluation of Eq (12b). If the Cartesian coordinate axes are

chosen to be parallel to the sides of the box, then on each face the normal derivativas become

simply the derivatives with respect to x, y or z. Sut in several cases (e.g. when P and Q lie

on the same vertical face) , these terms are zero. Where the derivatives are non-zero, they are

trivial to obtain from G. The second scurce of efficiency lies in the possibility of making

exteBsive use of explicit integration on for the computation of matrices U and V in Eq (12).

The series and integrai general forms of Green's function for deep and intermediate water

depths, given by John (1950), are discussed in Z. The particular intermediate depth forms used

in obtaining the results below are given inS. Nhenever possible, the series form is used for

the real part of G and its derivatives. These are explicitly integrated in the vertical direction,

but numerical integration is used for horizontal separation between points on S3. The series

is poorly convergent, however, when the horizontal separation between the points Pend Q is small, (and divergent when P and Q coincide) so that resort must then be made to the integral form. The

imaginary part of G, common to both series and integral forms, is integrated explicitly, except for horizontal integrations when points P and Q do not lie on the same vertical face. The real

part of the integral form contains a principal value integral and conventional source and image

terms. Both the latter are integrated explicitly in space, with the advantage that certain

singularities are avoided. The contribution of the principal value integral to matrices and '/

in Eq (12) is evaluated by a reversal of the order of integration after the singularity has been

smoothed. This is advantageous both when explicit integration may be used on S3 and when

numeri-cal integration is necessary. We have in fact used the explicit approach for all vertical

inte-grations (thereby avoiding another singularity at the free surface), and for horizontal

integra-tions when P and Q lie on the same vertical face.

In addition to these numerical features, the other major advantage in specifying Sj as a

rectangular box is the control afforded on irregular frequencies. This phenomenon, which Only

arises for surface piercing bodies, is associated with non-unique solutions of the integral

equation, Eq (10) , at certain discrete frequencies corresponding to the eigenvalues of a

homo-;genous problem posed in V1. Specifically, the problem as discussed by John (1950) concerns

harmonic

functions

satisfying the free surface condition and a Dirichlet condition on S. If

is a rectangular box, of plan dimensions 2a x 2b and draught h, the appropriate functions may readily be shown to be - mr x = sin - (- - 1) nr - 1) sinh (z + h) on 2 a

siny-

b on u K coth K h = mn mn mn g K2 mr 2 _nr 2 mn + (-) , m, n = 1,2,

Thus for a given box geometry, the irregular frequencies a may be predicted from Eq (17). If

such a frequency lies within the r-ange of practical intere, another geometry may immediately

be selected before the hydrodynamic analysis is attempted. This may not even be always necessary,

however, if only certain modes of motion are of interest. It may be shown that the potentials

associated with symmetric modes of a symmetric body (e.g. heave, sway and roll for a body symmet-ric about Oyz) are only affected at those frequencies corresponding to eigenfunctions mn

possessing similar symmetries. Thus for example the horizontal force and overturning moment on

a fixed offshore structure would be unaffected by the lowest irregular frequency. The Use of Multiple Fluid Regions

When the coupled element method as described was applied to the analysis of multi-bodies

(e.g. twin pontoon and multi-cylinder systems), it was found to be extremely versatile from the

point of view of element mesh arrangement in the region around and betwee t1e various bodies.

A fine mesh may be used close to the bodies, and this may be graded into a coarser mesh further

-away. This has the advantage that relatively few nodes are placed on the boundary S5 and the

boundary integral part of the computation is accordingly limited in complexity. Nevertheless,

the direct analysis of multi-bodies was not considered to meet the required standards of economy,

and the irregular frequency problem could not be as easily avoided as in a small region V1 around a single body.

The mesh fineness in the element region is governed by the criterion that the greatest horizontal dimension of an element in any direction should generally be no greater than half the

-shortest wavelength in the flow (although it is shown in Z that this may be relaxed for certain

bodies and for certain depths of element submergence). ma large region enclosing more than

one body, this criterion leads to what we regard as an unduly large number of nodes. There is

no practical difficulty in making such a computation, since the programs (through secondary disc storage) permit virtually unlimited element regions, and the matrices generally remain well

conditioned. The cost, however, is considerably higher than when the coupled method is applied

to single bodies. Furthermore, the irregular frequencies become progressively lower as the

dimensions of the boundary S, are increased (c.f. Eq (17) and (18)). aecause of the close spacing

of these frequencies, and the fact that - except for symmetric geometries - the potentials

associated with both symmetric and anti-symmetric modes of motion are affected at each irregu-larityt the phenomenon becomes awkward to avoid in this direct approach to multi-body analysis.

Fortunately, it is possible to overcome all of the above difficulties. The coupled finite

element boundary integral method has sufficient flexibility to permit the use of a separate

boundary S, and associated element region v, around each body (where 8 is the identifier of

each particular body). The mathematical formu_lation summarised above is unchanged if such local

boundaries are joined by vanishingly thin vertical boundary segments. Integrals in opposite

directions over the latter segments cancel, so that there is no need to model these thin segments

or indeed to consider them further. The local boundaries S may be bro9ht very close to the

individual bodies, and only a few elements need be used in each region V . Thus practical

diff-culties associated with analysis of short wave length flows are curtaile. In addition, irregular frequencies are easily predicted and avoided, in a similar manner to that described for a single

boundary. Thus, for example, if two identical bodies and associated boundaries are placed

symmetrically about the y-z plane, the irregular frequencies are the same as those pertaining

to each body individually. The only difference is that both symmetric and anti-symmetric modes

are affected at all of the irregular frequencies. Examples are given among the results below.

where

Exploitation of Symmetry

The efficiency of the coupled element method may be further dramatically increased for bodies having one or two planes of symmetry. When this feature is exploited, it may not always

be necessary to invoke the use of separate boundaries S for multi-bodies - consider for example

an array of four vertical cylinders symmetrically placee in close proximity. Alternatively,

symmetry may be exploited in such a case, but the boundary brought close to the body instead of bounding one quadrant of a single, larger, inner region. In the analyses described below,

symmetry is exploited wherever possible, in addition to the use of multiple boundaries.

Let us first consider the finite element region itself, and the case of a single plane of

symmetry (y = 0, say) . The nodal potentials satisfy Eq (12) , which for convenience we write in

the abbreviated form

(19)

We define submatrices associated with the plane of symmetry (subscript Y) , and with the two

regions on either side of that plane (subscripts A and B for nodes with positive and negative

y coordinates respectively). Equation (19) may therefore be written in the partitioned form

430 AA

KT

- AY K K --AY A5 AY

2YY

AY T K A3 AY -AA

where K is defined as containing contributions only from elements one side of y = O (e.g. in

region

XL

For potentials symmetric about y = O (e.g. radiation potentials for heave, surge and

pitch)

A = and Eq (20) may be condensed to

(K

+K

) K

-AA -AB -AY

YY A

p

+p

-YA -YB

?YA

For potentials antisymmetric about y = O (e.g. radiation potentials for sway, roll and yaw),

= - and = 0; in that case Eq (20) becomes

AA - AB A = (22)

(20)

(21)

We note that in Eq (21) and (22) only has contributions from the boundary integral terms on S3

i.e. from K)2) in Eq (13): the matrices for the finite element region itself are of narrow band

form. Hence we may form the combinations (KAA + _{AB) and )AA -} AB) by exploiting the symmetry

and antisymmetry properties of the Green's functions themselves, as discussed below.

The diffraction potential for an arbitrary wave direction is of course neither symmetric

nor antisymmetric about y = 0. It may however be written as a sum of the symmetric component

+

) with the term , and the antisymmetric component

(7)

( , so that Eq (21)

and Eq (22) may still be usedwith appropriate definitions for the various terms.

If there are no planes of symmetry, the three dimensional radiation-diffraction analysis normally requires solution of a system of equations having seven right hand sides (i.e. n in

Eq (14c( takes the value nr where r 1 to 7). If however one plane of symmetry is exploited, we ay solve two smaller systems, each having four right hand sides: the symmetric system has right hand sides derived from nj, n, n5 and + n75), while for the antisymmetric system we

use n, n, n

and (n.

- n).

If there are two planes of symmetry this procedure may of course

be further extended: te original system of equations is reduced to four smaller sets as follows. Let the four quadrants, defined by the planes of symmetry, be designated A, B, C and D. The

right hand sides of the four sets of equations are then based on

n and (n;A + n7 _{+ n;C + n;D) for set 1;}

n1, n and )n.A + n73 _- - n75) for set 2;

n, n' and )n.A _- - n + n75) for set 3;

n and n - n' + n' - nL ) for set 4.

These correspond respectively to potentials symmetric about both planes; symmetric about y O

but antisvmmetric about x = O; antisymmetric about y = O but symmetric about x = O; and

anti-symmetric about both clanes. The diffraction potentials in the four quadrants are then determined

from the solutions associated with the ith set of equations (i = i to 4) using

= - (* + + (7) 1 (7) (7) (7) (7) 4 1 + - *3 - *4 (7) (7) (7) (7) I - ¡ - *2 - *3 + *4 (7) i (7) (7) (7) (7) = 4 1 - + *3 - *4

Finally we consider exploitation of symmetry in the discretised boundary integral equations,

Eq (i2a) and (12b) (c.f. discussion of KAB following Eq (22)). If there are two planes we defi.ne

four basic Greens functions G(A, ) G(5, )

, G(,

) and G(p0, q): here , p and

identify the images of _A in the other three quadrants. The functions may conveniently be called

'GA, G2, GC and GD.

Combinations

of these basic functions then yield Greens functions having

the appropriate symmetry and antisyr.metry properties, in a manner entirely analogous to the

previously discussed diffraction potentials. Thus, for example, the function G2 _{(GA+GB_Gc_GD)}

implies antisymmetry about y = O and symmetry about x = O; the three other combinations and

associated symmetry properties are equivalent to those specified for n.A, n.3, etc. These

properties may be verified by noting that G2 = O on y = O and 2G2/2x O on x = O, with similar

relations for other three cases. The integrals of Eq (12) are therefore reduced to integrals

over only one quarter of (associated with quadrant A). When there are two planes of symmetry

these must be evaluated for the four functions GA G3, G and GD, but the required number of terms

is still one quarter of those needed when no symmetry exists. This important saving is in

addition to that associated with reducing by a factor of four the order of boundary equations

to be solved (Stated formally in Eq (15)). Computational Aspects

(23a)

(23h)

(2 3c)

(23f)

Brief reference is made here to the related matters of array storage and equation solution.

Further details ars given in Z. Cur implementation of the coupled element approach is closely

tied to use of a frontal formulation and solution of the governing matrix equations. An essential

requirement of the algorithm is that sequential numbers of adjacent elements should not differ by a large amount; but the numbering of nodes, and

consequently

the bandwidth of the coefficient

matrix, is irrelevant. This has permitted the use of a much more general mesh generator than

would otherwise have been possible. The most important features of the algorithm areas follows:

processing of one element at a timer retention in core of only those coefficients which are partially complete; storage on disc of those coefficients which have been factorised and are needed for back substitution; and solution by back substitution for the unknowns at nodes whose coefficients were complete at the point of processing each element. Crout's factorisation is

used, and symmetry of the coefficient matrices is exploited wherever possible.

The elements are ordered such that, after the first stage of factorisation, only

coeffi-cients associated with nodes on the finite element boundary S remain in core. Up to this stage all arithmetic may be ir. real mode, but once the boundary terms are added the arithmetic becomes

complex. The final elimination solves for nodal potentials on S3 followed by back substitution

for nodes within the finite element region.

he maximum front width occurring during factorisation for the element region is largely governed by the number of nodes on the boundary S. This number is also the order of the boundary integral matrix equations. It istherefore highly desirable to select graduated element meshes for complex problems, such that while regions of interest or importance are finely discretised,

regions far from the body - especially those adjacent to - are more coarsely represented.

Should the front width nevertheless become so karge that the front matrix can no longer be accom-modated in core, then the matrix may be stored on a random access disc file and accessed one row ata time. While this greatly increases the capacity of the computer program, a severe penalty

is paid because of the high overhead in data transfer. Present experience indicates that, on

a medium sized computer, secondary storage of the front matrix will not usually be required. When for example our three dimensional program is implemented on a CDC 7600 computer - with 26k

words small core memory and 87k words large core memory - a front width of 400 is permissible

without resort to secondary storage of the front matrix, Provided that this value is not exceeded,

then any number of nodes may be generated within the finite element region. (The arrays are

currently dimensioned for 3000 unknowns). Under these circumstances, no special provision of

secondary storage is normally required in the factorisation of the boundary integral terms,

this context, it should be noted that matrix U in Eq (15) is not symmetric.

Finally, since hydrodynamic analyses are very often required for the same body at several different frequencies, it is important to isolate those sections of the analysis which ace inde-pendent of frecuency, and to perform them only once. Examples are computations of the addresses

of coefficients employed in the frontal scheme, and evaluations of frequency independent integrals. Both these secs of information may be stored on disc, provided that the number of data transfer operations thereby involved is not more expensive than re-computation cf the various quantities. Similarly it may be desirable automatically to renumber elements, so that those which are not adjacent to frequency dependent boundaries need only be processed once; although it is important

that the front width be not greatly increased as a result.

APPLICATIONS TO STANDARD MULTI-BODY CONFIGURATIONS

Three Vertical Cylinders

To demonstrate the resolution of complex flows past multi-component structures, seve-ral

examples are discussed in this and the following sections. Because of the inherent interest in

wave interaction effects between adjacent bodies, attention is restricted to systems floating

at the free surface.

The first example concetns the three vertical circular cylinders shown in Fig 2, for which exact analytical solutions to the diffraction problem have been obtained by Ohkusu (1975). Our

first coupled element analysis was based on the use of a single inner region V1 and boundary Sj,

with one plane of symmetry. A typical arrangement of elements (mesh 1) having 413 nodes is shown

in Fig 3, and results for the sway and heave exciting forces are given in Fig 4. The forces have

been non-dimensionalised as shown, with A as the wave amplitude, and the dimensionless frequency scale is the product of wave number k with cylinder radius a. There is seen to be fair agreement

between the analytical solution and the numerical results from mesh I at low frequency, as might

be anticipated on the basis of the excellent performance of the coupled element method for single

bodies as shown in B. When however the wavelength tends to the dimension of the largest element

used, the numerical solutions become increasingly inaccurate. Even with a finer mesh, having

571 nodes such that the maximum element size was halved, the results were quite inaccurate above

ka = 1.2. This is due principally to the phenomenon of irregular frequencies. For the boundary

surrounding mesh i and its finer derivative, mesh 2, the first twelve such frequencies occur n

the range from ka = 0.518 to ka = 1.584. The lower frequencies are well separated, and accurate

results may be obtained except in their very close vicinity (see also the corresponding discussion

for two dimensional problems in A). The higher frequencies, however,beccme progressively closer,

and it becomes impossible to obtain results which are uninfluenced by the poor matrix conditioning

which arises very near an irregular frequency. Further numerical studies have clearly shown that this problem is not ailleviated by choice of a different single boundary enclosing all three

cylinders.

The alternative approach of separate boundaries around each cylinder was therefore adopted. Three degrees of mesh fineness were investigated (each incorporating one plane of symmetry) these were mesh 3 with nine elements, mesh 4 with eighteen elements (shown in Fig 5), and mesh 5 with

thirty six elements. The sway and heave wave forces obtained from meshes 3 and 4 are compared

with the previous results in Fig 4. Also included in this comparison are results obtained by

Matsui (1981), using the boundary integral formulation for multiple axisymmetric bodies described

by Matsui and Tamaki (1981). The coupled element results based on separate boundaries are seen

to agree satisfactorily with those of Ohkusu (1975) and Matsui (1981) throughout the frequency

range plotted. Furthermore, results from the finest discretisation, mesh 5, were found to lie

so close to those from mesh 4 as to suggest rapid convergence of the coupled element formulation

(the reasons for the differences from Ohkusu's results for F2 around ka = 0.2 - 0.3 are not clear,

although they may be due to the procedure we adopted for extracting Ohkusus data from a small published graph - the coupled element values agree with those of Matsui)

Further comparisons of results for the three cylinder problem are given in Fig 6, showing

non-dimensional sway damping coefficients. ..-Again the plotted values from mesh 4 are virtually

indistinguishable from results (not plotted) from mesh 5. Agreement with the results of Matsui

(1981) is good, but the damping coefficients of Ohkusu (1975) show a very different behaviour. This is probably explained by the fact that standing wave terms were neglected in Ohkusu's calculations of the damping coefficients.

The central processing (CPU) times for the coupled element solutions to the three cylinder

problem are given in Table 1. In this, N is the total number of nodes, ne is the total number

of elements, N is the number of nodes on the boundary S3 (it may be recalled that each element

face has eight nodes). T0 represents the time taken for the computations independent of frequency,

whereas TF is the time per frequency for the frequency dependent computations (including of course equations solution and evaluation of hydrodynamic coefficients). These times relate to use of the

program on the CDC 7600 computer of the University cf Lcndn Computer Centre. It is evident that

not only does the use of separate boundaries improve the accuracy of the solutions, it also

effects a major decrease in computer time. Two Norizontal Cylinders

In order that standing wave and three dimensional effects might be studied, the two hori-zontal floating circular cylinders shown in Fig 7 were analysed using both two and three

dimensional coupled element formulations. In the former case both infinite (d/5 = ) and finite

(0/a = 10= water depths were examined, using a mesh of eight elements around each semicircular

cross-section of radius a. (Analogous single cylinder problems have been discussed extensively

in A.) For the three dimensional study two lengths of cylinder were considered, respectively

5 and 10 times the radius, in finite water depth (d/5 = 10). Separate boundaries were used

around each cylinder, and two planes of symmetry were exploited using mesh 6 (131 nodes) shown

in Fig 8. The longer cylinders were also analysed by an alternative discretisation, mesh 7

(178 nodes), having a finer subdivision in the circumferential direction.

The heave added mass coefficients obtained from the numerical analyses are compared

In--Fig 9 with a two dimensional (20) exact analytical soltion for infinite depth, obtained by

Ohkusu (1969). The frequency scale is va, where y =

/.

The 2D infinite depth coupled element

results are in excellent agreement with the analytical solution; the 2D finite depth results

only differ from these at low frequencies, as would be expected. Three dimensional effects are

clearly demonstrated, particulaily of course for the shorter cylinders. These are most

signi-ficant in the range of frequencies near that at which the 20 added mass abruptly becomes negative

(as a result of standing wave effects). As the cylinder length is decreased the added mass

variation becomes less abrupt, and the frequency associated with the sign change increases.

Sxamination of the surface elevation between the cylinders, as predicted by the coupled element method, shows that standing wave effects are also present in the 3D problems; but their intensity decreases with reduction in cylinder length as a result of flow around the ends of the cylinders.

The maximum surface elevation is confirmed by the numerical analysis to lie in the plane bisec-ting the length of the cylinders.

The heave damping coefficients obtained numerically are shown in Fig 10. Results from

mesh 7 have been omitted from both this and Fig 9, because they are very close to those from

mesh 6. Our coupled element results are compared with the 2D numerical solution for infinite

depth obtained by Nordenstrom et al (1971), using the Frank Close Fit boundary integral procedre.

Agreement between the infinite depth results is very close. It is found that the 2D damping

has a sharp maximum at around va = 0.65, where the added mass passes through zero and the internal

wave between the cylinders has a maximum amplitude. The 20 damping then dips steeply towards

zero at around va = 0.82, the frequency at which the internal wave is found to be wholly in phase

with the imposed motions. Above this frequency, the radiation damping from the two infinite

cylinders is very small. The 3D results again display less abrupt changes, and the frequencies

corresponding to maximum and minimum damping increase as the cylinder length decreases. Above

va = 1.2, however, the damping even for the shorter cylinder becomes very small.

In the case of 2D flow, the graph of non-dimensional heave exciting force in beam seas

has a similar form to that of heave damping, as shown in Fig 11. This might be anticipated on

the basis of the direct theoretical relationships between 2D wave force and radiation damping,

as discussed by Newman (1977), but this similarity appeals less marked as three dimensional

effects become more pronounced. Bai (1981) observed a related phenomenon in a study of a ship

in a canal, and it is also found when sway damping is compared with horizontal exciting force

on the twin cylinders. The latter is plotted in Fig 12 in non-dimensionala form. In this case

much smoother behaviour is observed, although a gentle oscillation may be discerned in the

results for the 3D flows.

I

On the basis of the meshes employed for the coupled element analyses, and the discussion

given above, it may be predicted that in the range O va 2 irregular frequencies occur as

follows: for the 2D cases at ua = 1.557; for the longer cylinders (2b/a = 10) at a = 1.584 and

1.664; and for the shorter cylinders (Th/a = 5) at ya = 1.664 and 1.961. Figure 13 shows the

heave added mass plotted at enlarged scale o.wer the range 1.25 . va 2.0, from which the

existence of the irregularities is clearly apparent. It is important to note that it has been

poss-ible to distinguish these Purely numerical phenomena(which do not exist physically) from

the abrupt changes of behaviour associated with the real effect of standing waves between the

cylinders. The numerical irregularities may be avoided through a change of aspect ratio of the

two local finite element boundaries.

-The CPU times for the three dimensional analyses are given in Table 1. It should be

recalled that in the two cylinder analysis two planes of symmetry were used. Hence the

average time per frequency (Tp.) includes that required for solution of the equations four times.

In the case of the three cylinder problem, however, the number of nodes (N) provided a diacre-tisaticn of half rather than s quarter of the body.

APPLICATIONS TO MORE COMPLEX MULTI-COMPONENT BODIES

Floating Box and Cylnier

As a first example of a more complex geometry than the cylinder configurations discussed above, we consider a problem analysed previously by van Oortmerssen (1979, 1981) and Loken (1981). This also introduces the additional feature of two bodies oscillating independently, in close

proximity. Generalised wave forces with added mass and damping matrices are obtained,

corres-ponding to modes of each body moving in its rigid degrees of freedom with the other body held

stationary. Thus for two three dimensional bodies there are generally twelve such modes, and

the added mass and damping matrices are of order 12. These terms are effectively equivalent

to the various interaction coefficients defined by van Oortrnerssen (1979).

The geometry of the problem is shown in Fig 11, scaled to the dimensions of the model

tests described by van Oortmerssen. The three meshes employed in our coupled element ideaLisation

are illustrated iii Fig 15: one plane of symmetry was exploited. Results are presented only for

waves approaching parallel to the plane of symmetry (and propagating in the direction towards

the cylinder from box). In the folLowing, therefore, we do not consider roll sway and yaw motions

of either body. The other six modes are defined by:

1 for surge of cylinder

2 for heave of cylinder

3 for pitch of cylinder about its centre of floatation

4 for surge of box

5 for heave of box

6 for pitch of box about its centre of floatation.

The added mass and damping terms are made non-dimensional according to the scheme given in

Table 2, where 7(1) and 2) are the submerged volumes of cylinder and box respectively,

L1

= O.958m and L(2) = 1.014m (see Fig 14).

Complete results fcr the hydrodynamic loads and responses are given in Z, frcm which a

small selection is presented here. Figure 16 shows typical comparisons of added mass and damping

coefficients computed by the coupled element approach, with measured data and numerical solutions

of van Oortmerssen (1979). The latter employed s uniform source distribution boundary integral

scheme, using 92 panels on the cylinder and 104 on the box. Agreement between the various results

is found to be generally very good for all coefficients, the greatest discrepancies arising in

the heave added mass and damping for the box (Figs 16c, 16d) : the coupled element results appear

to be converging toe slightly different line from the boundary integral approximation. Such

differences are relatively irrelevant to the computation of first order responses, although they

are likely to have greater influence on drift forces.

T]e first order wave forces on the two bodies, estimated from mesh 9, are shown in Fig 17.

There are no published data with which to compare these. Responses of the freely floating bodies

in waves may, however be compared, albeit for a different water depth. The response tests

described by van Oortmerssen (1981) were conducted in a tank having 1m water depth. Coupled

element responses were therefore re-computed for this depth, and results for the six relevant

modes are given in Fig 18. Agreement between the various numerical and experimental data are

again fairly satisfactory, with the exception of the sharp peaks or troughs in the response curves predicted by the boundary integral calculation of van Oortmerssen (1981) at u = 0.3 rad/s. These

features result from coupling with pitch motions of the box, and may be shown to be very sensitive to the frequency of pitch resonance: results in Z suggest that a small change in this resonant frequency can cause a major change in the form of these peeks or troughs in the heave and surge

responses. Now the pitch rescoance predicted by van Oortmcrssen (1981) occurs at u = 0.30 rad/s,

whereas our coupled element results suggest the value u = 0.37 rad/s. (A suggested reason for

this discrepancy, based on a discussion of metacentric heights, is given in Z.) Hence t 15

not surprising that our results in Fig 18 do not predict the peaks and troughs near u = 0.3 rad/e, features which are also clearly absent fromhe experimental results.

We conclude this study of the floating box and cylinder with the following observations. Our coupled element results are generally in agreement with published and numerical data, end we have some indication of the convergenceof our results from the three meshes cons.dered. The

intermediate mesh, from which the responses plotted in Fig 18 were computed, was economical to use, requiring less than 30 secs CPU time per frequency on the CDC 7600 computer. (Times for

each of the meshes are given in Table 1.) The meshes themselves were also very easy to define.

The existence of two bodies moving independently gave rise to no additional complications in the definition of added mass and damping, since these matrices are defined in association with

pre-determined modes for both single and multi-body problems. Finally, we note the important

influence of hydrodynamic interaction between bodies: this is typified by the influence of the box

resonance in pitch on the responses of the cylinder. Van Oortmerssen (1981) has shown that such

interactions are even tore significant in the case of wave drift.

Semisubmer s ib le Catsoaran

As an illustration of use of the- coupled element method for analysing _{the motions of a}

characteristic offshore structure, our final example concerns the semisubmersible _{shown at model}

scale in Fig 19. _{This was rreviousiy analysed by Tassi et al (1970), using strip}

theory and the

long wave approximation for the diffraction potential, in _{water of infinite depth. Model tests}

results have also been published by Tasai et al _{(1970), based on experiments in a 3m deep wave}

tank. The linear damping coefficients used in these authors _{analysis were obtained from}

tran-sient decay tests, with the model oscillating freely in the tank. Thus their analysis took some

account of viscous effects, but not by means of the now conventional quadratic _{drag terms.}

Nevertheless, the results of this early analysis were in surprisingly _{good agreement with the}

model test data.

The mesh employed for the coupled element analysis (mesh _{11) is shown in Fig 20. Two} planes of symmetry and the multi-boundary facïlity _{were exploited for maximum efficiency, although}

this is not necessary. _{The resulting mesh characteristics are included in Table 1, again}

showing

a CPU time of about 30 secs per frequency for the complete hydrodynamic load _{and response analysis.}

Included in this time is the computation of detailed _{pressure distributions around the}

_coluths

and pontoons of the semisubmersible, and local fluid particle kinematics. _{Indeed full details}

of the flow field may readily be obtained from these results, _{as for example if the effect of}

additional Norison type slender members is to be included (e.g. Garrison et al (1975)).

Details of the comparisons will be presented elsehwere, space only permitting _{a brief}

summary herein. _{The coupled element prediction of heave added mass over the frequency}

range

O to 6.3 cad/s varied between 72% and 90% of the frequency _{independent value used by Tassi et al.}

The damping was entirely different. _{The vertical wave exciting force (in beam seas) agreed}

very

well between the two methods of calculation, except at low _{frequencies where finite depth effects}

are important. _{Small differences at higher frequencies are associated with the long wave}

approxi-mation used by Tassi et al, and the associated assumption that forces in phase with wave article

velocities are neglicible. _{The resulting heave response in beam seas is plotted against}

fre-quency in Fig 21. _{There is good agreement between the experimental data and the two theoretical}

solutions, except near the heave resonance: there the coupled element results seriously under-estimate the influence of damping, since viscous effects are ignored.

Comparison has also been made for the other modes, and fair agreement observed. Discre-pancies between the theoretical solutions arise principally from the infinite depth _assumption of Tasai et al, their neglect of certain coupling terms (e.g. pitch-surge), _{and the different}

treatments of damping. _{Differences between coupled element and experimental results are probably}

due mainly to viscous effects (being particularly noticeable _{near the various resonances) but}

these are likely to be less significant at full scale.

This investigation of the semisubrsersible catamaran has confirmed the viability of

employing the coupled element formulation for analysis of such structures. Although it may be

unnecessary to use three dimensional potential flow analysis for many semisubmersibe and tethered buoyant platform designs, others have particularly large volume pontoons giving rise to

signi-ficant 3D wave diffraction (but negligible viscoOs( effects. Where these have traditionally

been assessed by means of model tests, it may soon be possible to _{use an efficient numerical}

technique.

CONCLUS IONS

The foregoing range of examples has been selected in order to illustrate the _{ease with} which loads and responses of multicomponent bodies may be numerically predicted. Comparison of

our results with alternative analyses and experimental data have confirmed the accuracy of the coupld element formulation, even when a relatively coarse mesh is employed. And when finer

meshes are used, the computations do not appear to be unduly expensive, either in terms of

compu-ter time or central memory reqirements.

There are certain further features which, because of space limitations, we have omitted

from the detailed results and comments for e&ch example. Two of these are briefly summarised

here. _{They relate to additional information which is available over and above the hydrodynamic}

forces and responses in rigid body modes.

Firstly, let us recall the not infrequent requirement for hydrodynamic forces between

closely spaced bodies. A well known example is the splitting force between the pontoons of a

semisubmersible, for which methods of estimation have been compared by Mathisen and Carlsen

(1980). This may be obtained directly from the multi-body formulation described above, by an

appropriate specification of the modes. Thus the splitting force between two parallel pontoons

may be regarded as a generalised force, including diffraction and radiation effects. It is

associated with the mode in which the pontoons are given equal and opposite unit displacements .n

significant (due to a relatively flexible connection between the pontoons), then the generalised displacement associated with the mode should be retained in the ecuations of motion. Under these

circumstances the radiation potential for this mode is required, and is obtained directly from

the coupled element analysis. If structural dynamic effects are not important, it is not even

necessary to solve the boundary value problem for this radiation potential: the splitting force

is a quasistatic phenomenon, again given directly by the analysis.

The second type of information sometimes required concerna pressure distributions and fluid kinematics, including free surface profiles and particle velocities. These may be obtained

from any of the three dimensional boundary integral techniques currently available. Details of

pressure distributions, however, are often poorly resolved, and difficulties are caused by the

singularities. On the submerged surface of a body, for example, a uniform distribution of source

strength or velocity potential is usually assumed over a panel or facet; and this determines the

resolution of the pressure distributions obtained. In the coupled element method described above,

however, velocity potentials, pressures etc vary quadratically within an element, and there are no problems due to singularities or discontinuities between elements.

This latter feature suggests that the approach illustrated in this paper should provide a powerful method of evaluating second order wave drift forces, when these are obtained by direct integration of pressures on the submerged surface of the body or bodies in question. For mean

drift forces obtained indirectly, via the momentum conservation theorem, there is the possibility of performing the required integrations on the localised finite element outer boundary

instead of on the body. Since S is prescribed to be a rectangular box, and the potentia3.s on

vary quadratically between nodal values obtained from the first order calculation, the

integra-tions are trivial. It also appears that the coupled element formulation could be well suited

to computation of the second order potential contribution to slowly varying drift forces. These matters are currently under investigation.

ACKNOWLEDGEMENTS

The work described in this paper was supported by the Marine Technology Directorate of the Science and Engineering Research Council through Grant GR/B/6870.9 to the London Centre for

Marine Technology. We are also grateful to Dr. T. Matsui of Nagoya University, Japan, for his

provision of numerical results for the three cylinder problem.

-REFERENCES

SAI, K.J. and YEUNG, R.W., 1974, Numerical Solution to Free Surface Flow Problems", Proceedings,

Tenth Naval Hydrodynamics Symposium, Cambridge, Mass., U.S.A.

SAI, K.J., 1981, "A Localised Finite Element Method for Three Dimensional Ship Motion Problems", Proceedings of Third International Conference on Numerical Ship Hydrodynamics, Volume 4,

Paper VII - 1, Paris Office of Naval Research.

CHEN, ILS. and MEl, C.C., 1974, "Oscillations and Wave Forces in a Man-Made Narbour in the Open Sea", Proceedings, Tenth Naval Hydrodynamics Symposium, Cambridge, Mass., U.S.A.

EATOCK TAYLOR, R. and ZIETSMAN, J., 1981a, (A*) , 'A Comparison of Localised Finite Element

Formu-lations for Two Dimensional Wave Diffraction and Radiation Problem", International Journal for Numerical Methods in Engineering, Volume 17, p 1355-1384.

EATOCK TAYLOR, R. and ZIETSMAN, J., lgBlb,(B*), "Implementation of Coupled Element Formulations for Hydrodynamic Analysis in Three Dimensions", Pp 281-290 in Numerical Methods for Coupled Probíems, Edited by Hinton, E., Bettess, P. and Lewis, R.W., Swansea, U.K.: Pineridge Press.

EUVRARD, O., JAMI, A., MORICE, C. and OUSSET, Y., 1977, "Calcul Numérique des Oscillations d'un Navire Engendrées par la Houle", Journal de Mécanique, Volume 16, po 269-394.

GARRISON, C.J. , TORUM, A., IVERSON, C., LEIVSETH, S. and EBBESMEYER, C.C., 1974, "Wave Forces on

Large Volume Structures - a Comparison between Theory and Model Tests', Offshore Technology

Conference, Paper 2137, Houston, Texas.

ITTC, 1981, "Report of the Ocean Engineering Comrnittee",l6th International Towing Tank Conference, Leningrad, USSR.

JOSH, F., 1950, "On the Motion of Floating Bodies II", Communications in Pure and Applied

Mathematics, Volume 3, pp 45-101.

* A and B respectively are used in the text to identify these references.

LENOIR, M. and JAMI, A., 1978, "A Variational Formulation for Exterior Problems in Linear

Hydro-dynamics", Computer Methods in Aoplied Mechanics and Engineerinc, Volume 16, pp 341-359.

LOKEN, A.E., 1981, "Rydrodynamic Interaction between Several Floating Bodies of Arbitrary Form in

Waves', Proceedings International Symposium on Hydrodynamics in Ocean Engineering, Troridheim,

Norway, Volume 2, pp 745-779. Trondheim, Norway: Norwegian Institute of Technology.

MATEISEN, J. and CARLSEN, C.A., 1980, "Comparison of Calculation Methods for Wave Loads',

Proceedings of International Symposium on Ocean Engineering Ship Handling, Sweden, 1980, 7,

pp 1-23. Gothenburg, Sweden: Swedish Maritime Research Centre, SSPA.

MATSUI, T. and TAMAKI, T., 1981, "Hydrodynamic Interaction between Group of Vertical Axisymmetric Bodies Floating in Waves', Proceedings International Symposium on Hydrodynamics in Ocean

Engineering, Trondheim, Norway, Volume 2, pp 817-836. Trondheim, Norway: Norwegian Institute of

Technology.

MATSUI, T., 1981, Private Communication.

NEWMAN, J.N., 1977, Marine Hydrodynamics. Cambridge, Mass: MIT Press.

NOROENSTROM, N., FALTINSEN, O. and PEDERSEN, s., 1971, "Prediction of Wave Induced Motions and

Loads for Catamarans', Offshore Technology Conference, Paper 1418, Houston, Texas.

OEKUSU, M., 1969, "On the Heaving Motion of Two Circular Cylinders on the Surface of a Flutd" Reporta of Research Institute for Applied Mechanics, Kyushu University, Japan, Volume XVII,

pp 167-185.

OEXtJSU, M., 1975, "Hydrodynamic Forces on Multiple Cylinders in Waves", Proceedings of

Inter-national Symposium on The Dynamics of Marine Vehicles and Structures in Waves, University College London, 1974, Edited Bishop, R.E.D. and Price, W.G. London. England: InstitutiOn of Mechanical

Engineers.

OORTMERSSEN, G. VAN, 1979, "Hydrodynamic Interaction between Two Structures Floating in Waves", Proceedings, 2nd International Conference on the Behaviour of Off-Shore Structures, London,

England, 1979, Volume 1, Paper 26, pp 339-356. Cranfield, Bedford,England:BHRA Fluid Engineering.

OORTMERSSEN, G. VAN, 1981, "Some Hydrodynamic Aspects of Multi-body Systems", Proceedings

International Symposium on Hydrodynamics in Ocean Engineering, Trondheim, Norway, Volume 2,

pp 725-744. Trondheim, Norway: Norwegian Institute of Technology.

TASAI, F., ARAKAWA, H. end KURIHARA, M., 1970, "A Study of the Motions of a Semi Submersible

Catamaran Hull in Regular Waves", Reports of Research Institute for Arolied Mechanics, Kyushu

University, Japan, Volume XVII, p 9-32.

ZIENKIEWICZ, O.C., 1977, The Finite Element Method. London, England: McGraw Hill.

ZIENKIEWICZ, D.C., BETTESS, P. and KELLY, D.W, 1977, "The Finite Element Method for Determining

Fluid Loadings on Rigid Structures: Two and Three Dimensional Formulations", Numerical Methods tn

Offshore Engineering, (Eds. Zienkiewicz, D.C., Lewis, R. and Stagg, K.G.), London, England:Wiley.

ZIETSMAN, J., 1982, (Z*), Localized Finite Element Formulations for Ocean Wsve Diffraction and

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London.

* _{us used in the text to identify this reference.}

437

438 -V (2) a /L ¡g A.. = /(p 7(1)); JJ 22 Aik = jk' 7(2)); - _{(1) /g/L1} ) B. A.

/(o

7 JJ Jj a. = A ¡(p V jk jk (2) /g/L2)

Table i . Mesh characteristics and CPU tines

j = 1, 2 j i, 2 J k = 4, 5

j=k=4,5.

a k;

Table 2. _{Definition of non-dimensional frequency and hydrodynamic coefficients}

Mesh N

e nej T0/sec TF/sec

1 413 41 70 10 37.4 38.5 3 Vertical cylinders 2 571 63 98 16 70.2 95.6 (one plane 3 94 39 9 9 1.11 6.75 ° symme try) 4 172 72 18 18 3.21 27.9 5 272 72 35 18 10.7 35.9 2 Horizontal cylinders (two planes of symmetry) 5 7 131 178 55 75 14 20 14 20 3.57 6.50 23.8 47.3 Box + cylinder (one plane of symmetry) 8 9 10 30 196 276 54 82 115 12 20 30 12 20 30 1.83 4.04 7.96 13.1 28.6 58.5 Semisubmersible (two planes of symmetry) 11 215 -47 31 10 15.6 50.6

1.5 wave force 1.0 0.5 0 o

Fig 1. Regions and boundaries for the coupled element formulation.

V2

Fig 2. Three floating vertical cylinders

-2? d c

geometry (- = 5, - = 40, - = 3.65).

0.5 ka 1.0 15

Fig 4. Three vertical cylinders - sway and heave exciting forces (4 mesh 1; x mesh 3:

V mesh 4; 0 Matsui (1981); - Ohkuau (1975)).

a) Full mesh.

b) iidden lines removed.

Fig 3. Three vertical cylinders -mesh 1 (413 nodes)

Fig 5. Three vertical cylinders - mesh 4 (172 nodes).

+

1-o

X22 3ra c

440 0.5

Fig 6. Three vertical cylinders - sway damping coefficients (x mesh 3; V esh 4; ®Matsui (1981); Ohkusu (1975)).

IL,,

2a d

4.0Fig 7. Two horizontal floating cylinders

-¿P d

geometry ) = 3, - = 10, ).

r

2a 2a

2F

z

Fg S. Two horizontal cylinders - -4.0-mesh 6 (131 nodes)

Fig 9. Two horizontal cylinders _{- heave added mass} coefficients. (+ 2D, _a = ; X 2D, _a 10; V 3D, = 10, = 10; a a S 3D, = 10, = 5; _{OhJcusu (1969)).} a a 0.5 ka 1.0 1.5

2npa2 d 8.. 7-6 5 o A3 1 V 4 -X + z z 3_ 4 V 2.. _{- V} Iç,VVVV O O va ¶ 1.0 2.0 3.0

Fig 10. Two horizontal cylinders - heave damping Fig il. Two horizontal cylinders heave exciting force coefficients (20, = x 2D, ¡ = 10; )+ 20, = ; Y. 20, 10; V 3D, = 10 d 2h a a a V 3D, - 10, - = 10; 0 3D, - = 10, - = 5) 2h d 2h a a a a _{10; 3 3D, - = 10} - = 5). fi 8 pgAab 0.5-. o.4_ X

Fig 12. Two horizontal cylinders - sway exciting force

d d - d 1+ 20, - = ; x 20 - = 10; V iD, - 10, a a a d = 10; 0 3D, - 10; - = 5) a a a 1.5 1.0 1 . 0- 0.5-f3 8ogAab O Q

VV

vV Y o 2î6i 1.0

Fig 13. TwO horizontal cylinders - heave added mass

coefficients with irregular frequencies (legend as for Fig 9.).

¡ ; j I i 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 va

20

3.0 o 2 -- 3 va 0.8.. o

V

X V O O 0.3 0.2 0.1 o

442

Fig 14. Floating box and cylinder - geometry (dimensions in metres)

A, 0.4

w

a) added mass A44

1.2 0.8

00

0.8. 0.958 V Z Z

Z Z Z

T. 3 Z 5 53

IC A 0.8 1 6 2.4 c) added mass 344 0.8 B55 1.6 1.6 2.4 b) added damping 344 0.4 0.2 0.5 51 -0.5 w d) added damping 355 U i Z -) f)interaction coefficient

Fig 16. Floating box and cylinder - hydrodynamic coefficients for water depth/draft = 7.33 (+ mesh 8; x mesh 9; 9 mesh 10;

- Van Oortmerssen 1981). mesh B _{(130 nodes)} mesh B )196 nodes) mesh 10 (276 nodes) f Nm/rn 400 300 200 100W o

012 34 56

el (rad/sec) a) exciting forces o 3 Q y f T

3v

TTTvT

0

123 45

6 w/ (rad/sec) b) exciting moments

Fig 17. Floating box and cylinder - wave forces for water depth/draft 7.33

(mesh 9) ..., t.i-.

=

YB 3 1.097 (4 T 4 xc

rn

o w e) interaction coefficient A .f s

Fig 15. Floating box and cylinder-meshes.

100008000

-24

0 0 0.8 1.6

24

o 8 1.6

3 2 o (2) 4 3 (3) 2 o

- o

'L

2 3 4 5 Period/sec

Fig 21. Semisubmersible catamaran - heave reamonse er unit wave amplitude in beam seas ( mesh li;

experiment; - theory

Fig 18. Floating box and cylinder - responses ç)r(er unit wave amplitude (r = to 6 with pitch responses ç> and ç(6 in degrees/n) for water depth/draft ratio

3.33 (9 mesh 9; experiment; - theory (Van Oortmerssen 1981)). 'Ir 0.16 0.62 0.62 0.62 2.26 0.95 0.75 X 'Ir 3.00

Fig 19. Semisubmersible catamaran - geometry (dimensicns in metres).

Fig 20. Semisubnersible catamaran - mesh 11 (two planes of symmetry, 215 nodes)

0 5 10 0 5 10 4 3 L 0 10 46) (4 ç 6 5 4 3 2 6 V 3 2 0 5 iO 0 5 10 0 5 10